On the representation theory of \(\pi\)-separable groups
type: Item
instance of: scholarly article
title: On the representation theory of \(\pi\)-separable groups
zbMATH Open document ID: 0656.20016
DOI: 10.1016/0021-8693(88)90086-5
author: Geoffrey R. Robinson
author: Reiner Staszewski
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: The paper under review gives a new and simplified approach to results of \textit{I. M. Isaacs} [J. Algebra 86, 98-128 (1984; Zbl 0526.20006)] on characters of \(\pi\)-separable groups where \(\pi\) is a set of primes. More specifically, the authors prove that, for any \(\pi\)-separable group, the group of generalized characters vanishing on \(\pi\)-singular elements has a basis such that every character vanishing on \(\pi\)-singular elements can be written as a non-negative linear combination of basis elements. They also show that the group of generalized characters restricted to \(\pi\)-regular elements has a basis such that the restriction of every character to \(\pi\)-regular elements can be written as a non-negative linear combination of basis elements. This generalizes results of P. Fong for the case where \(\pi\) consists of a single prime. In the proof the authors make use of the representation theory of finite groups over certain semilocal rings.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 4071180
zbMATH Keywords: Krull-Schmidt theorem
zbMATH Keywords: projective module
zbMATH Keywords: characters of \(\pi\)-separable groups
zbMATH Keywords: group of generalized characters
zbMATH Keywords: \(\pi\)-singular elements
zbMATH Keywords: \(\pi\)- regular elements
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90086-5
OpenAlex ID: W2053739454
cites work: On The Characters of p-Solvable Groups
cites work: Correspondences of Characters for Relatively Prime Operator Groups
cites work: Characters of \(\pi\)-separable groups
cites work: Characters of Relatively Projective Modules II
cites work: On Characters of Relatively Projective Modules
description: scientific article; zbMATH DE number 4071180